*4.2. Graphical Illustrations*

This section is dedicated to the numerical results, their validation, and the discussion. To examine the inclusion of all the leading parameters numerically, computational software MATLAB is used for the numerical simulations. Table 1 is drawn for the computed convergen<sup>t</sup> outcomes of *Nux*/*Re*1/2 *x* , *Shx*/*Re*1/2 *x* , and *Nnx*/*Re*1/2 *x* across the number of collocation points *N*, *Nt*, and *Nb* by fixing other parameters, whereas, Table 2 depicts the comparability of −θ(0) with previously published data [53–55] across *Pr* with the preceding investigations by fixing other parameters of the governing equations. Table 3 is calculated to compare our computational results with the shooting method, and it can be observe that the results matched perfectly with the shooting method results. Figures 1–11 have been plotted against all the leading parameters for microorganism distribution, nanoparticle concentration, temperature, and velocity distribution, respectively.

**Table 1.** Numerical convergen<sup>t</sup> values of Nusselt number, Sherwood Number, and the local density number of the motile microorganisms across *N*, *Nt*, and *Nb* by fixing *M* = 1, β*D* = *Ec* = 0, *Nr* = 0.5, *Rb* = 0.5, *Gr R*2*e*= 0.5, *Pr* = 10, *Le* = 10, *Lb* = 2, *Pe* = 0.5, Ω*d* = 1.0.


**Table 2.** Comparison of the current outcomes for Nusselt number with the previous investigations.


**Table 3.** Comparison of the present method with shooting technique.


**Figure 2.** Variation of β*D* and *M* on velocity distribution. Black line: *M* = 0, Red line: *M* = 3.

**Figure 3.** Variation of *Nr* and *Gr R*2*e*on velocity distribution. Black line: *Gr R*2*e*= 5.5, Red line: *Gr R*2*e*= 10.5.

**Figure 4.** Variation of *Rb* and *Gr R*2*e*on velocity distribution. Black line: *Gr R*2*e*= 5.5, Red line: *Gr R*2*e*= 10.5.

**Figure 5.** Variation of *Pr* and *M* on temperature profile. Black line: *M* = 0, Red line: *M* = 3.

**Figure 6.** Variation of *Nb* and *Nt* on temperature profile. Black line: *Nb* = 0.2, Red line: *Nb* = 0.5.

**Figure 7.** Variation of *Nb* and *Ec* on temperature profile. Black line: *Nb* = 0.2, Red line: *Nb* = 0.5.

**Figure 8.** Variation of *Nt* and *Le* on concentration profile. Solid line: *Nt* = 5, Dotted line: *Nt* = 10.

**Figure 9.** Variation of *Nb* and *Le* on concentration profile. Solid line: *Le* = 1.5, Dotted line: *Le* = 2.0.

**Figure 10.** Variation of *Pe* and *Lb* on microorganism profile. Solid line: *Pe* = 0.5, Dotted line: *Pe* = 2.0.

**Figure 11.** Variation of *Pe* and Ω*d* on microorganism profile. Solid line: Ω*d* = 0.1, Dotted line: Ω*d* = 0.3.

Figure 2 shows that the velocity distribution decelerates by enhancing the permeability parameter β*<sup>D</sup>*. It can be seen as a deceleration in momentum by taking increment in *M*, due to the existing body-force brought through the magnetic field. A well-known Lorentz force, causing a decrement for the velocity overshooting and momentum boundary-layer thickness. In Figure 3, it is recorded that by taking the increment in *Nr*, the velocity distribution decreases as a result of an increase in the negate buoyancy generated through the existence of nanoparticles, while for the Richardson number *Gr*/*R*2*e*, it is also found to be decreased by enhancing the values of the Richardson number. Figure 4 portrays that, by taking an increment in *Rb*, the velocity distribution falls because the power of convection due to bioconvection boosted against the convection of buoyancy force. In contrast, for the Richardson number *Gr*/*R*2*e*, it is found to be decreased by enlarging the values of the Richardson number.

The influence of Prandtl number *Pr*, Hartmann number *M*, the Brownian-motion parameter *Nb*, the thermophoresis parameter *Nt*, local Eckert number *Ec*, for various numeric values are drawn through Figures 5–7. From Figure 5, it is determined that by taking an increment in Prandtl number *Pr*, the temperature distribution slows down, although by enhancing the Hartmann number *M*, it accelerates the temperature distribution. Figure 6 is adorned for the effect of thermophoresis parameter *Nt* and the Brownian-motion parameter *Nb* of the temperature distribution, and also notice that the temperature distribution boosts for both parameters by enhancing the numeric value of these parameters. The influence of Eckert number *Ec* and the Brownian-motion parameter *Nb* of the temperature distribution is sketched in Figure 7, and it is noticed that the temperature distribution boosts for both parameters by enhancing the numeric value of these parameters. The further heating due to the interacting of the fluid to nanoparticles because of the Brownian-motion, thermophoresis impact, and viscous dissipation enhance the temperature. Therefore, the thickness of the thermal boundary layer turns into high-thicker across the larger numeric of *Nt*, *Nb*, and *Ec*, and temperature overshoots into the neighborhood of the stretched permeable sheet.

The impact of bioconvection Lewis number *Le*, the Brownian-motion parameter *Nb*, the thermophoresis parameter *Nt*, the bioconvection *Lb*, Peclet number *Pe*, and the microorganisms concentration difference parameter Ω*d* for concentration distribution and the density of motile microorganisms successively are shown through Figures 8–11. Figure 8 is adorned for the effect of bioconvection Lewis number *Le* and thermophoresis parameter *Nt* of the concentration distribution, and also observed that the concentration distribution decelerates by enhancing the numeric value of Lewis number *Le* because the convection of nanoparticles enhances by adding more immense value in Lewis number *Le*, and also found decremented by taking increment in thermophoresis parameter *Nt*. Therefore, the nanoparticles' boundary layer thickening has been developed to grow thicker with *Nt*. From Figure 9, it observed that by enlarging the Brownian-motion parameter *Nb* and the bioconvection Lewis number *Le*, the concentration profile slows down for both the parameters. The graphical behavior of various values of the bioconvection *Lb* and Peclet number *Pe* in Figure 10 portrays that a decrement in the density for motile microorganisms quickly occurs by enhancing the bioconvection *Lb* and Peclet number *Pe*. That is, the density of motile microorganisms sharply slow downed, and indeed, by strengthening the bioconvection Lewis number *Lb* and Peclet number *Pe*, interprets the decrement of microorganisms diffusion, hence the density and boundary layer thickness together downturns for motile microorganisms by rising value in *Lb* and *Pe*. The influence of the Peclet number *Pe* and the concentration of the microorganisms varying parametric quantity Ω*d* is sketched in Figure 11, and it is found that the density of motile microorganisms slowed down by enhancing both the parameters, i.e., the Peclet number *Pe* and the concentration of the microorganisms varying parametric quantity Ω*d*.
